"""
Procedures for fitting marginal regression models to dependent data
using Generalized Estimating Equations.

References
----------
KY Liang and S Zeger. "Longitudinal data analysis using
generalized linear models". Biometrika (1986) 73 (1): 13-22.

S Zeger and KY Liang. "Longitudinal Data Analysis for Discrete and
Continuous Outcomes". Biometrics Vol. 42, No. 1 (Mar., 1986),
pp. 121-130

A Rotnitzky and NP Jewell (1990). "Hypothesis testing of regression
parameters in semiparametric generalized linear models for cluster
correlated data", Biometrika, 77, 485-497.

Xu Guo and Wei Pan (2002). "Small sample performance of the score
test in GEE".
http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf

LA Mancl LA, TA DeRouen (2001). A covariance estimator for GEE with
improved small-sample properties.  Biometrics. 2001 Mar;57(1):126-34.
"""
from statsmodels.compat.python import lzip
from statsmodels.compat.pandas import Appender

import numpy as np
from scipy import stats
import pandas as pd
import patsy
from collections import defaultdict
from statsmodels.tools.decorators import cache_readonly
import statsmodels.base.model as base
# used for wrapper:
import statsmodels.regression.linear_model as lm
import statsmodels.base.wrapper as wrap

from statsmodels.genmod import families
from statsmodels.genmod.generalized_linear_model import GLM, GLMResults
from statsmodels.genmod import cov_struct as cov_structs

import statsmodels.genmod.families.varfuncs as varfuncs
from statsmodels.genmod.families.links import Link

from statsmodels.tools.sm_exceptions import (ConvergenceWarning,
                                             DomainWarning,
                                             IterationLimitWarning,
                                             ValueWarning)
import warnings

from statsmodels.graphics._regressionplots_doc import (
    _plot_added_variable_doc,
    _plot_partial_residuals_doc,
    _plot_ceres_residuals_doc)
from statsmodels.discrete.discrete_margins import (
    _get_margeff_exog, _check_margeff_args, _effects_at, margeff_cov_with_se,
    _check_at_is_all, _transform_names, _check_discrete_args,
    _get_dummy_index, _get_count_index)


class ParameterConstraint:
    """
    A class for managing linear equality constraints for a parameter
    vector.
    """

    def __init__(self, lhs, rhs, exog):
        """
        Parameters
        ----------
        lhs : ndarray
           A q x p matrix which is the left hand side of the
           constraint lhs * param = rhs.  The number of constraints is
           q >= 1 and p is the dimension of the parameter vector.
        rhs : ndarray
          A 1-dimensional vector of length q which is the right hand
          side of the constraint equation.
        exog : ndarray
          The n x p exognenous data for the full model.
        """

        # In case a row or column vector is passed (patsy linear
        # constraints passes a column vector).
        rhs = np.atleast_1d(rhs.squeeze())

        if rhs.ndim > 1:
            raise ValueError("The right hand side of the constraint "
                             "must be a vector.")

        if len(rhs) != lhs.shape[0]:
            raise ValueError("The number of rows of the left hand "
                             "side constraint matrix L must equal "
                             "the length of the right hand side "
                             "constraint vector R.")

        self.lhs = lhs
        self.rhs = rhs

        # The columns of lhs0 are an orthogonal basis for the
        # orthogonal complement to row(lhs), the columns of lhs1 are
        # an orthogonal basis for row(lhs).  The columns of lhsf =
        # [lhs0, lhs1] are mutually orthogonal.
        lhs_u, lhs_s, lhs_vt = np.linalg.svd(lhs.T, full_matrices=1)
        self.lhs0 = lhs_u[:, len(lhs_s):]
        self.lhs1 = lhs_u[:, 0:len(lhs_s)]
        self.lhsf = np.hstack((self.lhs0, self.lhs1))

        # param0 is one solution to the underdetermined system
        # L * param = R.
        self.param0 = np.dot(self.lhs1, np.dot(lhs_vt, self.rhs) /
                             lhs_s)

        self._offset_increment = np.dot(exog, self.param0)

        self.orig_exog = exog
        self.exog_fulltrans = np.dot(exog, self.lhsf)

    def offset_increment(self):
        """
        Returns a vector that should be added to the offset vector to
        accommodate the constraint.

        Parameters
        ----------
        exog : array_like
           The exogeneous data for the model.
        """

        return self._offset_increment

    def reduced_exog(self):
        """
        Returns a linearly transformed exog matrix whose columns span
        the constrained model space.

        Parameters
        ----------
        exog : array_like
           The exogeneous data for the model.
        """
        return self.exog_fulltrans[:, 0:self.lhs0.shape[1]]

    def restore_exog(self):
        """
        Returns the full exog matrix before it was reduced to
        satisfy the constraint.
        """
        return self.orig_exog

    def unpack_param(self, params):
        """
        Converts the parameter vector `params` from reduced to full
        coordinates.
        """

        return self.param0 + np.dot(self.lhs0, params)

    def unpack_cov(self, bcov):
        """
        Converts the covariance matrix `bcov` from reduced to full
        coordinates.
        """

        return np.dot(self.lhs0, np.dot(bcov, self.lhs0.T))


_gee_init_doc = """
    Marginal regression model fit using Generalized Estimating Equations.

    GEE can be used to fit Generalized Linear Models (GLMs) when the
    data have a grouped structure, and the observations are possibly
    correlated within groups but not between groups.

    Parameters
    ----------
    endog : array_like
        1d array of endogenous values (i.e. responses, outcomes,
        dependent variables, or 'Y' values).
    exog : array_like
        2d array of exogeneous values (i.e. covariates, predictors,
        independent variables, regressors, or 'X' values). A `nobs x
        k` array where `nobs` is the number of observations and `k` is
        the number of regressors. An intercept is not included by
        default and should be added by the user. See
        `statsmodels.tools.add_constant`.
    groups : array_like
        A 1d array of length `nobs` containing the group labels.
    time : array_like
        A 2d array of time (or other index) values, used by some
        dependence structures to define similarity relationships among
        observations within a cluster.
    family : family class instance
%(family_doc)s
    cov_struct : CovStruct class instance
        The default is Independence.  To specify an exchangeable
        structure use cov_struct = Exchangeable().  See
        statsmodels.genmod.cov_struct.CovStruct for more
        information.
    offset : array_like
        An offset to be included in the fit.  If provided, must be
        an array whose length is the number of rows in exog.
    dep_data : array_like
        Additional data passed to the dependence structure.
    constraint : (ndarray, ndarray)
        If provided, the constraint is a tuple (L, R) such that the
        model parameters are estimated under the constraint L *
        param = R, where L is a q x p matrix and R is a
        q-dimensional vector.  If constraint is provided, a score
        test is performed to compare the constrained model to the
        unconstrained model.
    update_dep : bool
        If true, the dependence parameters are optimized, otherwise
        they are held fixed at their starting values.
    weights : array_like
        An array of case weights to use in the analysis.
    %(extra_params)s

    See Also
    --------
    statsmodels.genmod.families.family
    :ref:`families`
    :ref:`links`

    Notes
    -----
    Only the following combinations make sense for family and link ::

                   + ident log logit probit cloglog pow opow nbinom loglog logc
      Gaussian     |   x    x                        x
      inv Gaussian |   x    x                        x
      binomial     |   x    x    x     x       x     x    x           x      x
      Poisson      |   x    x                        x
      neg binomial |   x    x                        x          x
      gamma        |   x    x                        x

    Not all of these link functions are currently available.

    Endog and exog are references so that if the data they refer
    to are already arrays and these arrays are changed, endog and
    exog will change.

    The "robust" covariance type is the standard "sandwich estimator"
    (e.g. Liang and Zeger (1986)).  It is the default here and in most
    other packages.  The "naive" estimator gives smaller standard
    errors, but is only correct if the working correlation structure
    is correctly specified.  The "bias reduced" estimator of Mancl and
    DeRouen (Biometrics, 2001) reduces the downward bias of the robust
    estimator.

    The robust covariance provided here follows Liang and Zeger (1986)
    and agrees with R's gee implementation.  To obtain the robust
    standard errors reported in Stata, multiply by sqrt(N / (N - g)),
    where N is the total sample size, and g is the average group size.
    %(notes)s
    Examples
    --------
    %(example)s
"""

_gee_nointercept = """
    The nominal and ordinal GEE models should not have an intercept
    (either implicit or explicit).  Use "0 + " in a formula to
    suppress the intercept.
"""

_gee_family_doc = """\
        The default is Gaussian.  To specify the binomial
        distribution use `family=sm.families.Binomial()`. Each family
        can take a link instance as an argument.  See
        statsmodels.genmod.families.family for more information."""

_gee_ordinal_family_doc = """\
        The only family supported is `Binomial`.  The default `Logit`
        link may be replaced with `probit` if desired."""

_gee_nominal_family_doc = """\
        The default value `None` uses a multinomial logit family
        specifically designed for use with GEE.  Setting this
        argument to a non-default value is not currently supported."""

_gee_fit_doc = """
    Fits a marginal regression model using generalized estimating
    equations (GEE).

    Parameters
    ----------
    maxiter : int
        The maximum number of iterations
    ctol : float
        The convergence criterion for stopping the Gauss-Seidel
        iterations
    start_params : array_like
        A vector of starting values for the regression
        coefficients.  If None, a default is chosen.
    params_niter : int
        The number of Gauss-Seidel updates of the mean structure
        parameters that take place prior to each update of the
        dependence structure.
    first_dep_update : int
        No dependence structure updates occur before this
        iteration number.
    cov_type : str
        One of "robust", "naive", or "bias_reduced".
    ddof_scale : scalar or None
        The scale parameter is estimated as the sum of squared
        Pearson residuals divided by `N - ddof_scale`, where N
        is the total sample size.  If `ddof_scale` is None, the
        number of covariates (including an intercept if present)
        is used.
    scaling_factor : scalar
        The estimated covariance of the parameter estimates is
        scaled by this value.  Default is 1, Stata uses N / (N - g),
        where N is the total sample size and g is the average group
        size.
    scale : str or float, optional
        `scale` can be None, 'X2', or a float
        If a float, its value is used as the scale parameter.
        The default value is None, which uses `X2` (Pearson's
        chi-square) for Gamma, Gaussian, and Inverse Gaussian.
        The default is 1 for the Binomial and Poisson families.

    Returns
    -------
    An instance of the GEEResults class or subclass

    Notes
    -----
    If convergence difficulties occur, increase the values of
    `first_dep_update` and/or `params_niter`.  Setting
    `first_dep_update` to a greater value (e.g. ~10-20) causes the
    algorithm to move close to the GLM solution before attempting
    to identify the dependence structure.

    For the Gaussian family, there is no benefit to setting
    `params_niter` to a value greater than 1, since the mean
    structure parameters converge in one step.
"""

_gee_results_doc = """
    Attributes
    ----------

    cov_params_default : ndarray
        default covariance of the parameter estimates. Is chosen among one
        of the following three based on `cov_type`
    cov_robust : ndarray
        covariance of the parameter estimates that is robust
    cov_naive : ndarray
        covariance of the parameter estimates that is not robust to
        correlation or variance misspecification
    cov_robust_bc : ndarray
        covariance of the parameter estimates that is robust and bias
        reduced
    converged : bool
        indicator for convergence of the optimization.
        True if the norm of the score is smaller than a threshold
    cov_type : str
        string indicating whether a "robust", "naive" or "bias_reduced"
        covariance is used as default
    fit_history : dict
        Contains information about the iterations.
    fittedvalues : ndarray
        Linear predicted values for the fitted model.
        dot(exog, params)
    model : class instance
        Pointer to GEE model instance that called `fit`.
    normalized_cov_params : ndarray
        See GEE docstring
    params : ndarray
        The coefficients of the fitted model.  Note that
        interpretation of the coefficients often depends on the
        distribution family and the data.
    scale : float
        The estimate of the scale / dispersion for the model fit.
        See GEE.fit for more information.
    score_norm : float
        norm of the score at the end of the iterative estimation.
    bse : ndarray
        The standard errors of the fitted GEE parameters.
"""

_gee_example = """
    Logistic regression with autoregressive working dependence:

    >>> import statsmodels.api as sm
    >>> family = sm.families.Binomial()
    >>> va = sm.cov_struct.Autoregressive()
    >>> model = sm.GEE(endog, exog, group, family=family, cov_struct=va)
    >>> result = model.fit()
    >>> print(result.summary())

    Use formulas to fit a Poisson GLM with independent working
    dependence:

    >>> import statsmodels.api as sm
    >>> fam = sm.families.Poisson()
    >>> ind = sm.cov_struct.Independence()
    >>> model = sm.GEE.from_formula("y ~ age + trt + base", "subject",
                                 data, cov_struct=ind, family=fam)
    >>> result = model.fit()
    >>> print(result.summary())

    Equivalent, using the formula API:

    >>> import statsmodels.api as sm
    >>> import statsmodels.formula.api as smf
    >>> fam = sm.families.Poisson()
    >>> ind = sm.cov_struct.Independence()
    >>> model = smf.gee("y ~ age + trt + base", "subject",
                    data, cov_struct=ind, family=fam)
    >>> result = model.fit()
    >>> print(result.summary())
"""

_gee_ordinal_example = """
    Fit an ordinal regression model using GEE, with "global
    odds ratio" dependence:

    >>> import statsmodels.api as sm
    >>> gor = sm.cov_struct.GlobalOddsRatio("ordinal")
    >>> model = sm.OrdinalGEE(endog, exog, groups, cov_struct=gor)
    >>> result = model.fit()
    >>> print(result.summary())

    Using formulas:

    >>> import statsmodels.formula.api as smf
    >>> model = smf.ordinal_gee("y ~ 0 + x1 + x2", groups, data,
                                    cov_struct=gor)
    >>> result = model.fit()
    >>> print(result.summary())
"""

_gee_nominal_example = """
    Fit a nominal regression model using GEE:

    >>> import statsmodels.api as sm
    >>> import statsmodels.formula.api as smf
    >>> gor = sm.cov_struct.GlobalOddsRatio("nominal")
    >>> model = sm.NominalGEE(endog, exog, groups, cov_struct=gor)
    >>> result = model.fit()
    >>> print(result.summary())

    Using formulas:

    >>> import statsmodels.api as sm
    >>> model = sm.NominalGEE.from_formula("y ~ 0 + x1 + x2", groups,
                     data, cov_struct=gor)
    >>> result = model.fit()
    >>> print(result.summary())

    Using the formula API:

    >>> import statsmodels.formula.api as smf
    >>> model = smf.nominal_gee("y ~ 0 + x1 + x2", groups, data,
                                cov_struct=gor)
    >>> result = model.fit()
    >>> print(result.summary())
"""


def _check_args(endog, exog, groups, time, offset, exposure):

    if endog.size != exog.shape[0]:
        raise ValueError("Leading dimension of 'exog' should match "
                         "length of 'endog'")

    if groups.size != endog.size:
        raise ValueError("'groups' and 'endog' should have the same size")

    if time is not None and (time.size != endog.size):
        raise ValueError("'time' and 'endog' should have the same size")

    if offset is not None and (offset.size != endog.size):
        raise ValueError("'offset and 'endog' should have the same size")

    if exposure is not None and (exposure.size != endog.size):
        raise ValueError("'exposure' and 'endog' should have the same size")


class GEE(GLM):

    __doc__ = (
        "    Marginal Regression Model using Generalized Estimating "
        "Equations.\n" + _gee_init_doc %
        {'extra_params': base._missing_param_doc,
         'family_doc': _gee_family_doc,
         'example': _gee_example,
         'notes': ""})

    cached_means = None

    def __init__(self, endog, exog, groups, time=None, family=None,
                 cov_struct=None, missing='none', offset=None,
                 exposure=None, dep_data=None, constraint=None,
                 update_dep=True, weights=None, **kwargs):

        if type(self) is GEE:
            self._check_kwargs(kwargs)
        if family is not None:
            if not isinstance(family.link, tuple(family.safe_links)):
                msg = ("The {0} link function does not respect the "
                       "domain of the {1} family.")
                warnings.warn(msg.format(family.link.__class__.__name__,
                                         family.__class__.__name__),
                              DomainWarning)

        groups = np.asarray(groups)  # in case groups is pandas

        if "missing_idx" in kwargs and kwargs["missing_idx"] is not None:
            # If here, we are entering from super.from_formula; missing
            # has already been dropped from endog and exog, but not from
            # the other variables.
            ii = ~kwargs["missing_idx"]
            groups = groups[ii]
            if time is not None:
                time = time[ii]
            if offset is not None:
                offset = offset[ii]
            if exposure is not None:
                exposure = exposure[ii]
            del kwargs["missing_idx"]

        self.missing = missing
        self.dep_data = dep_data
        self.constraint = constraint
        self.update_dep = update_dep

        self._fit_history = defaultdict(list)

        # Pass groups, time, offset, and dep_data so they are
        # processed for missing data along with endog and exog.
        # Calling super creates self.exog, self.endog, etc. as
        # ndarrays and the original exog, endog, etc. are
        # self.data.endog, etc.
        super().__init__(endog, exog, groups=groups,
                         time=time, offset=offset,
                         exposure=exposure, weights=weights,
                         dep_data=dep_data, missing=missing,
                         family=family, **kwargs)

        _check_args(
            self.endog,
            self.exog,
            self.groups,
            self.time,
            getattr(self, "offset", None),
            getattr(self, "exposure", None),
        )

        self._init_keys.extend(["update_dep", "constraint", "family",
                                "cov_struct"])
        # remove keys added by super that are not supported
        try:
            self._init_keys.remove("freq_weights")
            self._init_keys.remove("var_weights")
        except ValueError:
            pass

        # Handle the family argument
        if family is None:
            family = families.Gaussian()
        else:
            if not issubclass(family.__class__, families.Family):
                raise ValueError("GEE: `family` must be a genmod "
                                 "family instance")
        self.family = family

        # Handle the cov_struct argument
        if cov_struct is None:
            cov_struct = cov_structs.Independence()
        else:
            if not issubclass(cov_struct.__class__, cov_structs.CovStruct):
                raise ValueError("GEE: `cov_struct` must be a genmod "
                                 "cov_struct instance")

        self.cov_struct = cov_struct

        # Handle the constraint
        self.constraint = None
        if constraint is not None:
            if len(constraint) != 2:
                raise ValueError("GEE: `constraint` must be a 2-tuple.")
            if constraint[0].shape[1] != self.exog.shape[1]:
                raise ValueError(
                    "GEE: the left hand side of the constraint must have "
                    "the same number of columns as the exog matrix.")
            self.constraint = ParameterConstraint(constraint[0],
                                                  constraint[1],
                                                  self.exog)

            if self._offset_exposure is not None:
                self._offset_exposure += self.constraint.offset_increment()
            else:
                self._offset_exposure = (
                    self.constraint.offset_increment().copy())
            self.exog = self.constraint.reduced_exog()

        # Create list of row indices for each group
        group_labels, ix = np.unique(self.groups, return_inverse=True)
        se = pd.Series(index=np.arange(len(ix)), dtype="int")
        gb = se.groupby(ix).groups
        dk = [(lb, np.asarray(gb[k])) for k, lb in enumerate(group_labels)]
        self.group_indices = dict(dk)
        self.group_labels = group_labels

        # Convert the data to the internal representation, which is a
        # list of arrays, corresponding to the groups.
        self.endog_li = self.cluster_list(self.endog)
        self.exog_li = self.cluster_list(self.exog)

        if self.weights is not None:
            self.weights_li = self.cluster_list(self.weights)

        self.num_group = len(self.endog_li)

        # Time defaults to a 1d grid with equal spacing
        if self.time is not None:
            if self.time.ndim == 1:
                self.time = self.time[:, None]
            self.time_li = self.cluster_list(self.time)
        else:
            self.time_li = \
                [np.arange(len(y), dtype=np.float64)[:, None]
                 for y in self.endog_li]
            self.time = np.concatenate(self.time_li)

        if (self._offset_exposure is None or
            (np.isscalar(self._offset_exposure) and
             self._offset_exposure == 0.)):
            self.offset_li = None
        else:
            self.offset_li = self.cluster_list(self._offset_exposure)
        if constraint is not None:
            self.constraint.exog_fulltrans_li = \
                self.cluster_list(self.constraint.exog_fulltrans)

        self.family = family

        self.cov_struct.initialize(self)

        # Total sample size
        group_ns = [len(y) for y in self.endog_li]
        self.nobs = sum(group_ns)
        # The following are column based, not on rank see #1928
        self.df_model = self.exog.shape[1] - 1  # assumes constant
        self.df_resid = self.nobs - self.exog.shape[1]

        # Skip the covariance updates if all groups have a single
        # observation (reduces to fitting a GLM).
        maxgroup = max([len(x) for x in self.endog_li])
        if maxgroup == 1:
            self.update_dep = False

    # Override to allow groups and time to be passed as variable
    # names.
    @classmethod
    def from_formula(cls, formula, groups, data, subset=None,
                     time=None, offset=None, exposure=None,
                     *args, **kwargs):
        """
        Create a GEE model instance from a formula and dataframe.

        Parameters
        ----------
        formula : str or generic Formula object
            The formula specifying the model
        groups : array_like or string
            Array of grouping labels.  If a string, this is the name
            of a variable in `data` that contains the grouping labels.
        data : array_like
            The data for the model.
        subset : array_like
            An array-like object of booleans, integers, or index
            values that indicate the subset of the data to used when
            fitting the model.
        time : array_like or string
            The time values, used for dependence structures involving
            distances between observations.  If a string, this is the
            name of a variable in `data` that contains the time
            values.
        offset : array_like or string
            The offset values, added to the linear predictor.  If a
            string, this is the name of a variable in `data` that
            contains the offset values.
        exposure : array_like or string
            The exposure values, only used if the link function is the
            logarithm function, in which case the log of `exposure`
            is added to the offset (if any).  If a string, this is the
            name of a variable in `data` that contains the offset
            values.
        {missing_param_doc}
        args : extra arguments
            These are passed to the model
        kwargs : extra keyword arguments
            These are passed to the model with two exceptions. `dep_data`
            is processed as described below.  The ``eval_env`` keyword is
            passed to patsy. It can be either a
            :class:`patsy:patsy.EvalEnvironment` object or an integer
            indicating the depth of the namespace to use. For example, the
            default ``eval_env=0`` uses the calling namespace.
            If you wish to use a "clean" environment set ``eval_env=-1``.

        Optional arguments
        ------------------
        dep_data : str or array_like
            Data used for estimating the dependence structure.  See
            specific dependence structure classes (e.g. Nested) for
            details.  If `dep_data` is a string, it is interpreted as
            a formula that is applied to `data`. If it is an array, it
            must be an array of strings corresponding to column names in
            `data`.  Otherwise it must be an array-like with the same
            number of rows as data.

        Returns
        -------
        model : GEE model instance

        Notes
        -----
        `data` must define __getitem__ with the keys in the formula
        terms args and kwargs are passed on to the model
        instantiation. E.g., a numpy structured or rec array, a
        dictionary, or a pandas DataFrame.
        """.format(missing_param_doc=base._missing_param_doc)

        groups_name = "Groups"
        if isinstance(groups, str):
            groups_name = groups
            groups = data[groups]

        if isinstance(time, str):
            time = data[time]

        if isinstance(offset, str):
            offset = data[offset]

        if isinstance(exposure, str):
            exposure = data[exposure]

        dep_data = kwargs.get("dep_data")
        dep_data_names = None
        if dep_data is not None:
            if isinstance(dep_data, str):
                dep_data = patsy.dmatrix(dep_data, data,
                                         return_type='dataframe')
                dep_data_names = dep_data.columns.tolist()
            else:
                dep_data_names = list(dep_data)
                dep_data = data[dep_data]
            kwargs["dep_data"] = np.asarray(dep_data)

        family = None
        if "family" in kwargs:
            family = kwargs["family"]
            del kwargs["family"]

        model = super().from_formula(formula, data=data, subset=subset,
                                     groups=groups, time=time,
                                     offset=offset,
                                     exposure=exposure,
                                     family=family,
                                     *args, **kwargs)

        if dep_data_names is not None:
            model._dep_data_names = dep_data_names
        model._groups_name = groups_name

        return model

    def cluster_list(self, array):
        """
        Returns `array` split into subarrays corresponding to the
        cluster structure.
        """

        if array.ndim == 1:
            return [np.array(array[self.group_indices[k]])
                    for k in self.group_labels]
        else:
            return [np.array(array[self.group_indices[k], :])
                    for k in self.group_labels]

    def compare_score_test(self, submodel):
        """
        Perform a score test for the given submodel against this model.

        Parameters
        ----------
        submodel : GEEResults instance
            A fitted GEE model that is a submodel of this model.

        Returns
        -------
        A dictionary with keys "statistic", "p-value", and "df",
        containing the score test statistic, its chi^2 p-value,
        and the degrees of freedom used to compute the p-value.

        Notes
        -----
        The score test can be performed without calling 'fit' on the
        larger model.  The provided submodel must be obtained from a
        fitted GEE.

        This method performs the same score test as can be obtained by
        fitting the GEE with a linear constraint and calling `score_test`
        on the results.

        References
        ----------
        Xu Guo and Wei Pan (2002). "Small sample performance of the score
        test in GEE".
        http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf
        """

        # Since the model has not been fit, its scaletype has not been
        # set.  So give it the scaletype of the submodel.
        self.scaletype = submodel.model.scaletype

        # Check consistency between model and submodel (not a comprehensive
        # check)
        submod = submodel.model
        if self.exog.shape[0] != submod.exog.shape[0]:
            msg = "Model and submodel have different numbers of cases."
            raise ValueError(msg)
        if self.exog.shape[1] == submod.exog.shape[1]:
            msg = "Model and submodel have the same number of variables"
            warnings.warn(msg)
        if not isinstance(self.family, type(submod.family)):
            msg = "Model and submodel have different GLM families."
            warnings.warn(msg)
        if not isinstance(self.cov_struct, type(submod.cov_struct)):
            warnings.warn("Model and submodel have different GEE covariance "
                          "structures.")
        if not np.equal(self.weights, submod.weights).all():
            msg = "Model and submodel should have the same weights."
            warnings.warn(msg)

        # Get the positions of the submodel variables in the
        # parent model
        qm, qc = _score_test_submodel(self, submodel.model)
        if qm is None:
            msg = "The provided model is not a submodel."
            raise ValueError(msg)

        # Embed the submodel params into a params vector for the
        # parent model
        params_ex = np.dot(qm, submodel.params)

        # Attempt to preserve the state of the parent model
        cov_struct_save = self.cov_struct
        import copy
        cached_means_save = copy.deepcopy(self.cached_means)

        # Get the score vector of the submodel params in
        # the parent model
        self.cov_struct = submodel.cov_struct
        self.update_cached_means(params_ex)
        _, score = self._update_mean_params()
        if score is None:
            msg = "Singular matrix encountered in GEE score test"
            warnings.warn(msg, ConvergenceWarning)
            return None

        if not hasattr(self, "ddof_scale"):
            self.ddof_scale = self.exog.shape[1]

        if not hasattr(self, "scaling_factor"):
            self.scaling_factor = 1

        _, ncov1, cmat = self._covmat()
        score2 = np.dot(qc.T, score)

        try:
            amat = np.linalg.inv(ncov1)
        except np.linalg.LinAlgError:
            amat = np.linalg.pinv(ncov1)

        bmat_11 = np.dot(qm.T, np.dot(cmat, qm))
        bmat_22 = np.dot(qc.T, np.dot(cmat, qc))
        bmat_12 = np.dot(qm.T, np.dot(cmat, qc))

        amat_11 = np.dot(qm.T, np.dot(amat, qm))
        amat_12 = np.dot(qm.T, np.dot(amat, qc))

        try:
            ab = np.linalg.solve(amat_11, bmat_12)
        except np.linalg.LinAlgError:
            ab = np.dot(np.linalg.pinv(amat_11), bmat_12)

        score_cov = bmat_22 - np.dot(amat_12.T, ab)

        try:
            aa = np.linalg.solve(amat_11, amat_12)
        except np.linalg.LinAlgError:
            aa = np.dot(np.linalg.pinv(amat_11), amat_12)

        score_cov -= np.dot(bmat_12.T, aa)

        try:
            ab = np.linalg.solve(amat_11, bmat_11)
        except np.linalg.LinAlgError:
            ab = np.dot(np.linalg.pinv(amat_11), bmat_11)

        try:
            aa = np.linalg.solve(amat_11, amat_12)
        except np.linalg.LinAlgError:
            aa = np.dot(np.linalg.pinv(amat_11), amat_12)

        score_cov += np.dot(amat_12.T, np.dot(ab, aa))

        # Attempt to restore state
        self.cov_struct = cov_struct_save
        self.cached_means = cached_means_save

        from scipy.stats.distributions import chi2
        try:
            sc2 = np.linalg.solve(score_cov, score2)
        except np.linalg.LinAlgError:
            sc2 = np.dot(np.linalg.pinv(score_cov), score2)
        score_statistic = np.dot(score2, sc2)
        score_df = len(score2)
        score_pvalue = 1 - chi2.cdf(score_statistic, score_df)
        return {"statistic": score_statistic,
                "df": score_df,
                "p-value": score_pvalue}

    def estimate_scale(self):
        """
        Estimate the dispersion/scale.
        """

        if self.scaletype is None:
            if isinstance(self.family, (families.Binomial, families.Poisson,
                                        families.NegativeBinomial,
                                        _Multinomial)):
                return 1.
        elif isinstance(self.scaletype, float):
            return np.array(self.scaletype)

        endog = self.endog_li
        cached_means = self.cached_means
        nobs = self.nobs
        varfunc = self.family.variance

        scale = 0.
        fsum = 0.
        for i in range(self.num_group):

            if len(endog[i]) == 0:
                continue

            expval, _ = cached_means[i]
            sdev = np.sqrt(varfunc(expval))
            resid = (endog[i] - expval) / sdev

            if self.weights is not None:
                f = self.weights_li[i]
                scale += np.sum(f * (resid ** 2))
                fsum += f.sum()
            else:
                scale += np.sum(resid ** 2)
                fsum += len(resid)

        scale /= (fsum * (nobs - self.ddof_scale) / float(nobs))

        return scale

    def mean_deriv(self, exog, lin_pred):
        """
        Derivative of the expected endog with respect to the parameters.

        Parameters
        ----------
        exog : array_like
           The exogeneous data at which the derivative is computed.
        lin_pred : array_like
           The values of the linear predictor.

        Returns
        -------
        The value of the derivative of the expected endog with respect
        to the parameter vector.

        Notes
        -----
        If there is an offset or exposure, it should be added to
        `lin_pred` prior to calling this function.
        """

        idl = self.family.link.inverse_deriv(lin_pred)
        dmat = exog * idl[:, None]
        return dmat

    def mean_deriv_exog(self, exog, params, offset_exposure=None):
        """
        Derivative of the expected endog with respect to exog.

        Parameters
        ----------
        exog : array_like
            Values of the independent variables at which the derivative
            is calculated.
        params : array_like
            Parameter values at which the derivative is calculated.
        offset_exposure : array_like, optional
            Combined offset and exposure.

        Returns
        -------
        The derivative of the expected endog with respect to exog.
        """

        lin_pred = np.dot(exog, params)
        if offset_exposure is not None:
            lin_pred += offset_exposure

        idl = self.family.link.inverse_deriv(lin_pred)
        dmat = np.outer(idl, params)
        return dmat

    def _update_mean_params(self):
        """
        Returns
        -------
        update : array_like
            The update vector such that params + update is the next
            iterate when solving the score equations.
        score : array_like
            The current value of the score equations, not
            incorporating the scale parameter.  If desired,
            multiply this vector by the scale parameter to
            incorporate the scale.
        """

        endog = self.endog_li
        exog = self.exog_li
        weights = getattr(self, "weights_li", None)

        cached_means = self.cached_means

        varfunc = self.family.variance

        bmat, score = 0, 0
        for i in range(self.num_group):

            expval, lpr = cached_means[i]
            resid = endog[i] - expval
            dmat = self.mean_deriv(exog[i], lpr)
            sdev = np.sqrt(varfunc(expval))

            if weights is not None:
                w = weights[i]
                wresid = resid * w
                wdmat = dmat * w[:, None]
            else:
                wresid = resid
                wdmat = dmat

            rslt = self.cov_struct.covariance_matrix_solve(
                    expval, i, sdev, (wdmat, wresid))
            if rslt is None:
                return None, None
            vinv_d, vinv_resid = tuple(rslt)

            bmat += np.dot(dmat.T, vinv_d)
            score += np.dot(dmat.T, vinv_resid)

        try:
            update = np.linalg.solve(bmat, score)
        except np.linalg.LinAlgError:
            update = np.dot(np.linalg.pinv(bmat), score)

        self._fit_history["cov_adjust"].append(
            self.cov_struct.cov_adjust)

        return update, score

    def update_cached_means(self, mean_params):
        """
        cached_means should always contain the most recent calculation
        of the group-wise mean vectors.  This function should be
        called every time the regression parameters are changed, to
        keep the cached means up to date.
        """

        endog = self.endog_li
        exog = self.exog_li
        offset = self.offset_li

        linkinv = self.family.link.inverse

        self.cached_means = []

        for i in range(self.num_group):

            if len(endog[i]) == 0:
                continue

            lpr = np.dot(exog[i], mean_params)
            if offset is not None:
                lpr += offset[i]
            expval = linkinv(lpr)

            self.cached_means.append((expval, lpr))

    def _covmat(self):
        """
        Returns the sampling covariance matrix of the regression
        parameters and related quantities.

        Returns
        -------
        cov_robust : array_like
           The robust, or sandwich estimate of the covariance, which
           is meaningful even if the working covariance structure is
           incorrectly specified.
        cov_naive : array_like
           The model-based estimate of the covariance, which is
           meaningful if the covariance structure is correctly
           specified.
        cmat : array_like
           The center matrix of the sandwich expression, used in
           obtaining score test results.
        """

        endog = self.endog_li
        exog = self.exog_li
        weights = getattr(self, "weights_li", None)
        varfunc = self.family.variance
        cached_means = self.cached_means

        # Calculate the naive (model-based) and robust (sandwich)
        # covariances.
        bmat, cmat = 0, 0
        for i in range(self.num_group):

            expval, lpr = cached_means[i]
            resid = endog[i] - expval
            dmat = self.mean_deriv(exog[i], lpr)
            sdev = np.sqrt(varfunc(expval))

            if weights is not None:
                w = weights[i]
                wresid = resid * w
                wdmat = dmat * w[:, None]
            else:
                wresid = resid
                wdmat = dmat

            rslt = self.cov_struct.covariance_matrix_solve(
                expval, i, sdev, (wdmat, wresid))
            if rslt is None:
                return None, None, None, None
            vinv_d, vinv_resid = tuple(rslt)

            bmat += np.dot(dmat.T, vinv_d)
            dvinv_resid = np.dot(dmat.T, vinv_resid)
            cmat += np.outer(dvinv_resid, dvinv_resid)

        scale = self.estimate_scale()

        try:
            bmati = np.linalg.inv(bmat)
        except np.linalg.LinAlgError:
            bmati = np.linalg.pinv(bmat)

        cov_naive = bmati * scale
        cov_robust = np.dot(bmati, np.dot(cmat, bmati))

        cov_naive *= self.scaling_factor
        cov_robust *= self.scaling_factor
        return cov_robust, cov_naive, cmat

    # Calculate the bias-corrected sandwich estimate of Mancl and
    # DeRouen.
    def _bc_covmat(self, cov_naive):

        cov_naive = cov_naive / self.scaling_factor
        endog = self.endog_li
        exog = self.exog_li
        varfunc = self.family.variance
        cached_means = self.cached_means
        scale = self.estimate_scale()

        bcm = 0
        for i in range(self.num_group):

            expval, lpr = cached_means[i]
            resid = endog[i] - expval
            dmat = self.mean_deriv(exog[i], lpr)
            sdev = np.sqrt(varfunc(expval))

            rslt = self.cov_struct.covariance_matrix_solve(
                expval, i, sdev, (dmat,))
            if rslt is None:
                return None
            vinv_d = rslt[0]
            vinv_d /= scale

            hmat = np.dot(vinv_d, cov_naive)
            hmat = np.dot(hmat, dmat.T).T

            f = self.weights_li[i] if self.weights is not None else 1.

            aresid = np.linalg.solve(np.eye(len(resid)) - hmat, resid)
            rslt = self.cov_struct.covariance_matrix_solve(
                expval, i, sdev, (aresid,))
            if rslt is None:
                return None
            srt = rslt[0]
            srt = f * np.dot(dmat.T, srt) / scale
            bcm += np.outer(srt, srt)

        cov_robust_bc = np.dot(cov_naive, np.dot(bcm, cov_naive))
        cov_robust_bc *= self.scaling_factor

        return cov_robust_bc

    def _starting_params(self):

        if np.isscalar(self._offset_exposure):
            offset = None
        else:
            offset = self._offset_exposure

        model = GLM(self.endog, self.exog, family=self.family,
                    offset=offset, freq_weights=self.weights)
        result = model.fit()
        return result.params

    @Appender(_gee_fit_doc)
    def fit(self, maxiter=60, ctol=1e-6, start_params=None,
            params_niter=1, first_dep_update=0,
            cov_type='robust', ddof_scale=None, scaling_factor=1.,
            scale=None):

        self.scaletype = scale

        # Subtract this number from the total sample size when
        # normalizing the scale parameter estimate.
        if ddof_scale is None:
            self.ddof_scale = self.exog.shape[1]
        else:
            if not ddof_scale >= 0:
                raise ValueError(
                    "ddof_scale must be a non-negative number or None")
            self.ddof_scale = ddof_scale

        self.scaling_factor = scaling_factor

        self._fit_history = defaultdict(list)

        if self.weights is not None and cov_type == 'naive':
            raise ValueError("when using weights, cov_type may not be naive")

        if start_params is None:
            mean_params = self._starting_params()
        else:
            start_params = np.asarray(start_params)
            mean_params = start_params.copy()

        self.update_cached_means(mean_params)

        del_params = -1.
        num_assoc_updates = 0
        for itr in range(maxiter):

            update, score = self._update_mean_params()
            if update is None:
                warnings.warn("Singular matrix encountered in GEE update",
                              ConvergenceWarning)
                break
            mean_params += update
            self.update_cached_means(mean_params)

            # L2 norm of the change in mean structure parameters at
            # this iteration.
            del_params = np.sqrt(np.sum(score ** 2))

            self._fit_history['params'].append(mean_params.copy())
            self._fit_history['score'].append(score)
            self._fit_history['dep_params'].append(
                self.cov_struct.dep_params)

            # Do not exit until the association parameters have been
            # updated at least once.
            if (del_params < ctol and
                    (num_assoc_updates > 0 or self.update_dep is False)):
                break

            # Update the dependence structure
            if (self.update_dep and (itr % params_niter) == 0
                    and (itr >= first_dep_update)):
                self._update_assoc(mean_params)
                num_assoc_updates += 1

        if del_params >= ctol:
            warnings.warn("Iteration limit reached prior to convergence",
                          IterationLimitWarning)

        if mean_params is None:
            warnings.warn("Unable to estimate GEE parameters.",
                          ConvergenceWarning)
            return None

        bcov, ncov, _ = self._covmat()
        if bcov is None:
            warnings.warn("Estimated covariance structure for GEE "
                          "estimates is singular", ConvergenceWarning)
            return None
        bc_cov = None
        if cov_type == "bias_reduced":
            bc_cov = self._bc_covmat(ncov)

        if self.constraint is not None:
            x = mean_params.copy()
            mean_params, bcov = self._handle_constraint(mean_params, bcov)
            if mean_params is None:
                warnings.warn("Unable to estimate constrained GEE "
                              "parameters.", ConvergenceWarning)
                return None

            y, ncov = self._handle_constraint(x, ncov)
            if y is None:
                warnings.warn("Unable to estimate constrained GEE "
                              "parameters.", ConvergenceWarning)
                return None

            if bc_cov is not None:
                y, bc_cov = self._handle_constraint(x, bc_cov)
                if x is None:
                    warnings.warn("Unable to estimate constrained GEE "
                                  "parameters.", ConvergenceWarning)
                    return None

        scale = self.estimate_scale()

        # kwargs to add to results instance, need to be available in __init__
        res_kwds = dict(cov_type=cov_type,
                        cov_robust=bcov,
                        cov_naive=ncov,
                        cov_robust_bc=bc_cov)

        # The superclass constructor will multiply the covariance
        # matrix argument bcov by scale, which we do not want, so we
        # divide bcov by the scale parameter here
        results = GEEResults(self, mean_params, bcov / scale, scale,
                             cov_type=cov_type, use_t=False,
                             attr_kwds=res_kwds)

        # attributes not needed during results__init__
        results.fit_history = self._fit_history
        self.fit_history = defaultdict(list)
        results.score_norm = del_params
        results.converged = (del_params < ctol)
        results.cov_struct = self.cov_struct
        results.params_niter = params_niter
        results.first_dep_update = first_dep_update
        results.ctol = ctol
        results.maxiter = maxiter

        # These will be copied over to subclasses when upgrading.
        results._props = ["cov_type", "use_t",
                          "cov_params_default", "cov_robust",
                          "cov_naive", "cov_robust_bc",
                          "fit_history",
                          "score_norm", "converged", "cov_struct",
                          "params_niter", "first_dep_update", "ctol",
                          "maxiter"]

        return GEEResultsWrapper(results)

    def _update_regularized(self, params, pen_wt, scad_param, eps):

        sn, hm = 0, 0

        for i in range(self.num_group):

            expval, _ = self.cached_means[i]
            resid = self.endog_li[i] - expval
            sdev = np.sqrt(self.family.variance(expval))

            ex = self.exog_li[i] * sdev[:, None]**2
            rslt = self.cov_struct.covariance_matrix_solve(
                           expval, i, sdev, (resid, ex))
            sn0 = rslt[0]
            sn += np.dot(ex.T, sn0)
            hm0 = rslt[1]
            hm += np.dot(ex.T, hm0)

        # Wang et al. divide sn here by num_group, but that
        # seems to be incorrect

        ap = np.abs(params)
        clipped = np.clip(scad_param * pen_wt - ap, 0, np.inf)
        en = pen_wt * clipped * (ap > pen_wt)
        en /= (scad_param - 1) * pen_wt
        en += pen_wt * (ap <= pen_wt)
        en /= eps + ap

        hm.flat[::hm.shape[0] + 1] += self.num_group * en
        sn -= self.num_group * en * params
        try:
            update = np.linalg.solve(hm, sn)
        except np.linalg.LinAlgError:
            update = np.dot(np.linalg.pinv(hm), sn)
            msg = "Encountered singularity in regularized GEE update"
            warnings.warn(msg)
        hm *= self.estimate_scale()

        return update, hm

    def _regularized_covmat(self, mean_params):

        self.update_cached_means(mean_params)

        ma = 0

        for i in range(self.num_group):

            expval, _ = self.cached_means[i]
            resid = self.endog_li[i] - expval
            sdev = np.sqrt(self.family.variance(expval))

            ex = self.exog_li[i] * sdev[:, None]**2
            rslt = self.cov_struct.covariance_matrix_solve(
                           expval, i, sdev, (resid,))
            ma0 = np.dot(ex.T, rslt[0])
            ma += np.outer(ma0, ma0)

        return ma

    def fit_regularized(self, pen_wt, scad_param=3.7, maxiter=100,
                        ddof_scale=None, update_assoc=5,
                        ctol=1e-5, ztol=1e-3, eps=1e-6, scale=None):
        """
        Regularized estimation for GEE.

        Parameters
        ----------
        pen_wt : float
            The penalty weight (a non-negative scalar).
        scad_param : float
            Non-negative scalar determining the shape of the Scad
            penalty.
        maxiter : int
            The maximum number of iterations.
        ddof_scale : int
            Value to subtract from `nobs` when calculating the
            denominator degrees of freedom for t-statistics, defaults
            to the number of columns in `exog`.
        update_assoc : int
            The dependence parameters are updated every `update_assoc`
            iterations of the mean structure parameter updates.
        ctol : float
            Convergence criterion, default is one order of magnitude
            smaller than proposed in section 3.1 of Wang et al.
        ztol : float
            Coefficients smaller than this value are treated as
            being zero, default is based on section 5 of Wang et al.
        eps : non-negative scalar
            Numerical constant, see section 3.2 of Wang et al.
        scale : float or string
            If a float, this value is used as the scale parameter.
            If "X2", the scale parameter is always estimated using
            Pearson's chi-square method (e.g. as in a quasi-Poisson
            analysis).  If None, the default approach for the family
            is used to estimate the scale parameter.

        Returns
        -------
        GEEResults instance.  Note that not all methods of the results
        class make sense when the model has been fit with regularization.

        Notes
        -----
        This implementation assumes that the link is canonical.

        References
        ----------
        Wang L, Zhou J, Qu A. (2012). Penalized generalized estimating
        equations for high-dimensional longitudinal data analysis.
        Biometrics. 2012 Jun;68(2):353-60.
        doi: 10.1111/j.1541-0420.2011.01678.x.
        https://www.ncbi.nlm.nih.gov/pubmed/21955051
        http://users.stat.umn.edu/~wangx346/research/GEE_selection.pdf
        """

        self.scaletype = scale

        mean_params = np.zeros(self.exog.shape[1])
        self.update_cached_means(mean_params)
        converged = False
        fit_history = defaultdict(list)

        # Subtract this number from the total sample size when
        # normalizing the scale parameter estimate.
        if ddof_scale is None:
            self.ddof_scale = self.exog.shape[1]
        else:
            if not ddof_scale >= 0:
                raise ValueError(
                    "ddof_scale must be a non-negative number or None")
            self.ddof_scale = ddof_scale

        # Keep this private for now.  In some cases the early steps are
        # very small so it seems necessary to ensure a certain minimum
        # number of iterations before testing for convergence.
        miniter = 20

        for itr in range(maxiter):

            update, hm = self._update_regularized(
                              mean_params, pen_wt, scad_param, eps)
            if update is None:
                msg = "Singular matrix encountered in regularized GEE update"
                warnings.warn(msg, ConvergenceWarning)
                break
            if itr > miniter and np.sqrt(np.sum(update**2)) < ctol:
                converged = True
                break
            mean_params += update
            fit_history['params'].append(mean_params.copy())
            self.update_cached_means(mean_params)

            if itr != 0 and (itr % update_assoc == 0):
                self._update_assoc(mean_params)

        if not converged:
            msg = "GEE.fit_regularized did not converge"
            warnings.warn(msg)

        mean_params[np.abs(mean_params) < ztol] = 0

        self._update_assoc(mean_params)
        ma = self._regularized_covmat(mean_params)
        cov = np.linalg.solve(hm, ma)
        cov = np.linalg.solve(hm, cov.T)

        # kwargs to add to results instance, need to be available in __init__
        res_kwds = dict(cov_type="robust", cov_robust=cov)

        scale = self.estimate_scale()
        rslt = GEEResults(self, mean_params, cov, scale,
                          regularized=True, attr_kwds=res_kwds)
        rslt.fit_history = fit_history

        return GEEResultsWrapper(rslt)

    def _handle_constraint(self, mean_params, bcov):
        """
        Expand the parameter estimate `mean_params` and covariance matrix
        `bcov` to the coordinate system of the unconstrained model.

        Parameters
        ----------
        mean_params : array_like
            A parameter vector estimate for the reduced model.
        bcov : array_like
            The covariance matrix of mean_params.

        Returns
        -------
        mean_params : array_like
            The input parameter vector mean_params, expanded to the
            coordinate system of the full model
        bcov : array_like
            The input covariance matrix bcov, expanded to the
            coordinate system of the full model
        """

        # The number of variables in the full model
        red_p = len(mean_params)
        full_p = self.constraint.lhs.shape[1]
        mean_params0 = np.r_[mean_params, np.zeros(full_p - red_p)]

        # Get the score vector under the full model.
        save_exog_li = self.exog_li
        self.exog_li = self.constraint.exog_fulltrans_li
        import copy
        save_cached_means = copy.deepcopy(self.cached_means)
        self.update_cached_means(mean_params0)
        _, score = self._update_mean_params()

        if score is None:
            warnings.warn("Singular matrix encountered in GEE score test",
                          ConvergenceWarning)
            return None, None

        _, ncov1, cmat = self._covmat()
        scale = self.estimate_scale()
        cmat = cmat / scale ** 2
        score2 = score[red_p:] / scale
        amat = np.linalg.inv(ncov1)

        bmat_11 = cmat[0:red_p, 0:red_p]
        bmat_22 = cmat[red_p:, red_p:]
        bmat_12 = cmat[0:red_p, red_p:]
        amat_11 = amat[0:red_p, 0:red_p]
        amat_12 = amat[0:red_p, red_p:]

        score_cov = bmat_22 - np.dot(amat_12.T,
                                     np.linalg.solve(amat_11, bmat_12))
        score_cov -= np.dot(bmat_12.T,
                            np.linalg.solve(amat_11, amat_12))
        score_cov += np.dot(amat_12.T,
                            np.dot(np.linalg.solve(amat_11, bmat_11),
                                   np.linalg.solve(amat_11, amat_12)))

        from scipy.stats.distributions import chi2
        score_statistic = np.dot(score2,
                                 np.linalg.solve(score_cov, score2))
        score_df = len(score2)
        score_pvalue = 1 - chi2.cdf(score_statistic, score_df)
        self.score_test_results = {"statistic": score_statistic,
                                   "df": score_df,
                                   "p-value": score_pvalue}

        mean_params = self.constraint.unpack_param(mean_params)
        bcov = self.constraint.unpack_cov(bcov)

        self.exog_li = save_exog_li
        self.cached_means = save_cached_means
        self.exog = self.constraint.restore_exog()

        return mean_params, bcov

    def _update_assoc(self, params):
        """
        Update the association parameters
        """

        self.cov_struct.update(params)

    def _derivative_exog(self, params, exog=None, transform='dydx',
                         dummy_idx=None, count_idx=None):
        """
        For computing marginal effects, returns dF(XB) / dX where F(.)
        is the fitted mean.

        transform can be 'dydx', 'dyex', 'eydx', or 'eyex'.

        Not all of these make sense in the presence of discrete regressors,
        but checks are done in the results in get_margeff.
        """
        # This form should be appropriate for group 1 probit, logit,
        # logistic, cloglog, heckprob, xtprobit.
        offset_exposure = None
        if exog is None:
            exog = self.exog
            offset_exposure = self._offset_exposure

        margeff = self.mean_deriv_exog(exog, params, offset_exposure)

        if 'ex' in transform:
            margeff *= exog
        if 'ey' in transform:
            margeff /= self.predict(params, exog)[:, None]
        if count_idx is not None:
            from statsmodels.discrete.discrete_margins import (
                _get_count_effects)
            margeff = _get_count_effects(margeff, exog, count_idx, transform,
                                         self, params)
        if dummy_idx is not None:
            from statsmodels.discrete.discrete_margins import (
                _get_dummy_effects)
            margeff = _get_dummy_effects(margeff, exog, dummy_idx, transform,
                                         self, params)
        return margeff

    def qic(self, params, scale, cov_params, n_step=1000):
        """
        Returns quasi-information criteria and quasi-likelihood values.

        Parameters
        ----------
        params : array_like
            The GEE estimates of the regression parameters.
        scale : scalar
            Estimated scale parameter
        cov_params : array_like
            An estimate of the covariance matrix for the
            model parameters.  Conventionally this is the robust
            covariance matrix.
        n_step : integer
            The number of points in the trapezoidal approximation
            to the quasi-likelihood function.

        Returns
        -------
        ql : scalar
            The quasi-likelihood value
        qic : scalar
            A QIC that can be used to compare the mean and covariance
            structures of the model.
        qicu : scalar
            A simplified QIC that can be used to compare mean structures
            but not covariance structures

        Notes
        -----
        The quasi-likelihood used here is obtained by numerically evaluating
        Wedderburn's integral representation of the quasi-likelihood function.
        This approach is valid for all families and  links.  Many other
        packages use analytical expressions for quasi-likelihoods that are
        valid in special cases where the link function is canonical.  These
        analytical expressions may omit additive constants that only depend
        on the data.  Therefore, the numerical values of our QL and QIC values
        will differ from the values reported by other packages.  However only
        the differences between two QIC values calculated for different models
        using the same data are meaningful.  Our QIC should produce the same
        QIC differences as other software.

        When using the QIC for models with unknown scale parameter, use a
        common estimate of the scale parameter for all models being compared.

        References
        ----------
        .. [*] W. Pan (2001).  Akaike's information criterion in generalized
               estimating equations.  Biometrics (57) 1.
        """

        varfunc = self.family.variance

        means = []
        omega = 0.0
        # omega^-1 is the model-based covariance assuming independence

        for i in range(self.num_group):
            expval, lpr = self.cached_means[i]
            means.append(expval)
            dmat = self.mean_deriv(self.exog_li[i], lpr)
            omega += np.dot(dmat.T, dmat) / scale

        means = np.concatenate(means)

        # The quasi-likelihood, use change of variables so the integration is
        # from -1 to 1.
        endog_li = np.concatenate(self.endog_li)
        du = means - endog_li
        qv = np.empty(n_step)
        xv = np.linspace(-0.99999, 1, n_step)
        for i, g in enumerate(xv):
            u = endog_li + (g + 1) * du / 2.0
            vu = varfunc(u)
            qv[i] = -np.sum(du**2 * (g + 1) / vu)
        qv /= (4 * scale)

        try:
            from scipy.integrate import trapezoid
        except ImportError:
            # Remove after minimum is SciPy 1.7
            from scipy.integrate import trapz as trapezoid
        ql = trapezoid(qv, dx=xv[1] - xv[0])

        qicu = -2 * ql + 2 * self.exog.shape[1]
        qic = -2 * ql + 2 * np.trace(np.dot(omega, cov_params))

        return ql, qic, qicu


class GEEResults(GLMResults):

    __doc__ = (
        "This class summarizes the fit of a marginal regression model "
        "using GEE.\n" + _gee_results_doc)

    def __init__(self, model, params, cov_params, scale,
                 cov_type='robust', use_t=False, regularized=False,
                 **kwds):

        super().__init__(
            model, params, normalized_cov_params=cov_params, scale=scale
        )

        # not added by super
        self.df_resid = model.df_resid
        self.df_model = model.df_model
        self.family = model.family

        attr_kwds = kwds.pop('attr_kwds', {})
        self.__dict__.update(attr_kwds)

        # we do not do this if the cov_type has already been set
        # subclasses can set it through attr_kwds
        if not (hasattr(self, 'cov_type') and
                hasattr(self, 'cov_params_default')):
            self.cov_type = cov_type  # keep alias
            covariance_type = self.cov_type.lower()
            allowed_covariances = ["robust", "naive", "bias_reduced"]
            if covariance_type not in allowed_covariances:
                msg = ("GEE: `cov_type` must be one of " +
                       ", ".join(allowed_covariances))
                raise ValueError(msg)

            if cov_type == "robust":
                cov = self.cov_robust
            elif cov_type == "naive":
                cov = self.cov_naive
            elif cov_type == "bias_reduced":
                cov = self.cov_robust_bc

            self.cov_params_default = cov
        else:
            if self.cov_type != cov_type:
                raise ValueError('cov_type in argument is different from '
                                 'already attached cov_type')

    @cache_readonly
    def resid(self):
        """
        The response residuals.
        """
        return self.resid_response

    def standard_errors(self, cov_type="robust"):
        """
        This is a convenience function that returns the standard
        errors for any covariance type.  The value of `bse` is the
        standard errors for whichever covariance type is specified as
        an argument to `fit` (defaults to "robust").

        Parameters
        ----------
        cov_type : str
            One of "robust", "naive", or "bias_reduced".  Determines
            the covariance used to compute standard errors.  Defaults
            to "robust".
        """

        # Check covariance_type
        covariance_type = cov_type.lower()
        allowed_covariances = ["robust", "naive", "bias_reduced"]
        if covariance_type not in allowed_covariances:
            msg = ("GEE: `covariance_type` must be one of " +
                   ", ".join(allowed_covariances))
            raise ValueError(msg)

        if covariance_type == "robust":
            return np.sqrt(np.diag(self.cov_robust))
        elif covariance_type == "naive":
            return np.sqrt(np.diag(self.cov_naive))
        elif covariance_type == "bias_reduced":
            if self.cov_robust_bc is None:
                raise ValueError(
                    "GEE: `bias_reduced` covariance not available")
            return np.sqrt(np.diag(self.cov_robust_bc))

    # Need to override to allow for different covariance types.
    @cache_readonly
    def bse(self):
        return self.standard_errors(self.cov_type)

    def score_test(self):
        """
        Return the results of a score test for a linear constraint.

        Returns
        -------
        Adictionary containing the p-value, the test statistic,
        and the degrees of freedom for the score test.

        Notes
        -----
        See also GEE.compare_score_test for an alternative way to perform
        a score test.  GEEResults.score_test is more general, in that it
        supports testing arbitrary linear equality constraints.   However
        GEE.compare_score_test might be easier to use when comparing
        two explicit models.

        References
        ----------
        Xu Guo and Wei Pan (2002). "Small sample performance of the score
        test in GEE".
        http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf
        """

        if not hasattr(self.model, "score_test_results"):
            msg = "score_test on results instance only available when "
            msg += " model was fit with constraints"
            raise ValueError(msg)

        return self.model.score_test_results

    @cache_readonly
    def resid_split(self):
        """
        Returns the residuals, the endogeneous data minus the fitted
        values from the model.  The residuals are returned as a list
        of arrays containing the residuals for each cluster.
        """
        sresid = []
        for v in self.model.group_labels:
            ii = self.model.group_indices[v]
            sresid.append(self.resid[ii])
        return sresid

    @cache_readonly
    def resid_centered(self):
        """
        Returns the residuals centered within each group.
        """
        cresid = self.resid.copy()
        for v in self.model.group_labels:
            ii = self.model.group_indices[v]
            cresid[ii] -= cresid[ii].mean()
        return cresid

    @cache_readonly
    def resid_centered_split(self):
        """
        Returns the residuals centered within each group.  The
        residuals are returned as a list of arrays containing the
        centered residuals for each cluster.
        """
        sresid = []
        for v in self.model.group_labels:
            ii = self.model.group_indices[v]
            sresid.append(self.centered_resid[ii])
        return sresid

    def qic(self, scale=None, n_step=1000):
        """
        Returns the QIC and QICu information criteria.

        See GEE.qic for documentation.
        """

        # It is easy to forget to set the scale parameter.  Sometimes
        # this is intentional, so we warn.
        if scale is None:
            warnings.warn("QIC values obtained using scale=None are not "
                          "appropriate for comparing models")

        if scale is None:
            scale = self.scale

        _, qic, qicu = self.model.qic(self.params, scale,
                                      self.cov_params(),
                                      n_step=n_step)

        return qic, qicu

    # FIXME: alias to be removed, temporary backwards compatibility
    split_resid = resid_split
    centered_resid = resid_centered
    split_centered_resid = resid_centered_split

    @Appender(_plot_added_variable_doc % {'extra_params_doc': ''})
    def plot_added_variable(self, focus_exog, resid_type=None,
                            use_glm_weights=True, fit_kwargs=None,
                            ax=None):

        from statsmodels.graphics.regressionplots import plot_added_variable

        fig = plot_added_variable(self, focus_exog,
                                  resid_type=resid_type,
                                  use_glm_weights=use_glm_weights,
                                  fit_kwargs=fit_kwargs, ax=ax)

        return fig

    @Appender(_plot_partial_residuals_doc % {'extra_params_doc': ''})
    def plot_partial_residuals(self, focus_exog, ax=None):

        from statsmodels.graphics.regressionplots import plot_partial_residuals

        return plot_partial_residuals(self, focus_exog, ax=ax)

    @Appender(_plot_ceres_residuals_doc % {'extra_params_doc': ''})
    def plot_ceres_residuals(self, focus_exog, frac=0.66, cond_means=None,
                             ax=None):

        from statsmodels.graphics.regressionplots import plot_ceres_residuals

        return plot_ceres_residuals(self, focus_exog, frac,
                                    cond_means=cond_means, ax=ax)

    def conf_int(self, alpha=.05, cols=None, cov_type=None):
        """
        Returns confidence intervals for the fitted parameters.

        Parameters
        ----------
        alpha : float, optional
             The `alpha` level for the confidence interval.  i.e., The
             default `alpha` = .05 returns a 95% confidence interval.
        cols : array_like, optional
             `cols` specifies which confidence intervals to return
        cov_type : str
             The covariance type used for computing standard errors;
             must be one of 'robust', 'naive', and 'bias reduced'.
             See `GEE` for details.

        Notes
        -----
        The confidence interval is based on the Gaussian distribution.
        """
        # super does not allow to specify cov_type and method is not
        # implemented,
        # FIXME: remove this method here
        if cov_type is None:
            bse = self.bse
        else:
            bse = self.standard_errors(cov_type=cov_type)
        params = self.params
        dist = stats.norm
        q = dist.ppf(1 - alpha / 2)

        if cols is None:
            lower = self.params - q * bse
            upper = self.params + q * bse
        else:
            cols = np.asarray(cols)
            lower = params[cols] - q * bse[cols]
            upper = params[cols] + q * bse[cols]
        return np.asarray(lzip(lower, upper))

    def summary(self, yname=None, xname=None, title=None, alpha=.05):
        """
        Summarize the GEE regression results

        Parameters
        ----------
        yname : str, optional
            Default is `y`
        xname : list[str], optional
            Names for the exogenous variables, default is `var_#` for ## in
            the number of regressors. Must match the number of parameters in
            the model
        title : str, optional
            Title for the top table. If not None, then this replaces
            the default title
        alpha : float
            significance level for the confidence intervals
        cov_type : str
            The covariance type used to compute the standard errors;
            one of 'robust' (the usual robust sandwich-type covariance
            estimate), 'naive' (ignores dependence), and 'bias
            reduced' (the Mancl/DeRouen estimate).

        Returns
        -------
        smry : Summary instance
            this holds the summary tables and text, which can be
            printed or converted to various output formats.

        See Also
        --------
        statsmodels.iolib.summary.Summary : class to hold summary results
        """

        top_left = [('Dep. Variable:', None),
                    ('Model:', None),
                    ('Method:', ['Generalized']),
                    ('', ['Estimating Equations']),
                    ('Family:', [self.model.family.__class__.__name__]),
                    ('Dependence structure:',
                     [self.model.cov_struct.__class__.__name__]),
                    ('Date:', None),
                    ('Covariance type: ', [self.cov_type, ])
                    ]

        NY = [len(y) for y in self.model.endog_li]

        top_right = [('No. Observations:', [sum(NY)]),
                     ('No. clusters:', [len(self.model.endog_li)]),
                     ('Min. cluster size:', [min(NY)]),
                     ('Max. cluster size:', [max(NY)]),
                     ('Mean cluster size:', ["%.1f" % np.mean(NY)]),
                     ('Num. iterations:', ['%d' %
                                           len(self.fit_history['params'])]),
                     ('Scale:', ["%.3f" % self.scale]),
                     ('Time:', None),
                     ]

        # The skew of the residuals
        skew1 = stats.skew(self.resid)
        kurt1 = stats.kurtosis(self.resid)
        skew2 = stats.skew(self.centered_resid)
        kurt2 = stats.kurtosis(self.centered_resid)

        diagn_left = [('Skew:', ["%12.4f" % skew1]),
                      ('Centered skew:', ["%12.4f" % skew2])]

        diagn_right = [('Kurtosis:', ["%12.4f" % kurt1]),
                       ('Centered kurtosis:', ["%12.4f" % kurt2])
                       ]

        if title is None:
            title = self.model.__class__.__name__ + ' ' +\
                "Regression Results"

        # Override the exog variable names if xname is provided as an
        # argument.
        if xname is None:
            xname = self.model.exog_names

        if yname is None:
            yname = self.model.endog_names

        # Create summary table instance
        from statsmodels.iolib.summary import Summary
        smry = Summary()
        smry.add_table_2cols(self, gleft=top_left, gright=top_right,
                             yname=yname, xname=xname,
                             title=title)
        smry.add_table_params(self, yname=yname, xname=xname,
                              alpha=alpha, use_t=False)
        smry.add_table_2cols(self, gleft=diagn_left,
                             gright=diagn_right, yname=yname,
                             xname=xname, title="")

        return smry

    def get_margeff(self, at='overall', method='dydx', atexog=None,
                    dummy=False, count=False):
        """Get marginal effects of the fitted model.

        Parameters
        ----------
        at : str, optional
            Options are:

            - 'overall', The average of the marginal effects at each
              observation.
            - 'mean', The marginal effects at the mean of each regressor.
            - 'median', The marginal effects at the median of each regressor.
            - 'zero', The marginal effects at zero for each regressor.
            - 'all', The marginal effects at each observation. If `at` is 'all'
              only margeff will be available.

            Note that if `exog` is specified, then marginal effects for all
            variables not specified by `exog` are calculated using the `at`
            option.
        method : str, optional
            Options are:

            - 'dydx' - dy/dx - No transformation is made and marginal effects
              are returned.  This is the default.
            - 'eyex' - estimate elasticities of variables in `exog` --
              d(lny)/d(lnx)
            - 'dyex' - estimate semi-elasticity -- dy/d(lnx)
            - 'eydx' - estimate semi-elasticity -- d(lny)/dx

            Note that tranformations are done after each observation is
            calculated.  Semi-elasticities for binary variables are computed
            using the midpoint method. 'dyex' and 'eyex' do not make sense
            for discrete variables.
        atexog : array_like, optional
            Optionally, you can provide the exogenous variables over which to
            get the marginal effects.  This should be a dictionary with the key
            as the zero-indexed column number and the value of the dictionary.
            Default is None for all independent variables less the constant.
        dummy : bool, optional
            If False, treats binary variables (if present) as continuous.  This
            is the default.  Else if True, treats binary variables as
            changing from 0 to 1.  Note that any variable that is either 0 or 1
            is treated as binary.  Each binary variable is treated separately
            for now.
        count : bool, optional
            If False, treats count variables (if present) as continuous.  This
            is the default.  Else if True, the marginal effect is the
            change in probabilities when each observation is increased by one.

        Returns
        -------
        effects : ndarray
            the marginal effect corresponding to the input options

        Notes
        -----
        When using after Poisson, returns the expected number of events
        per period, assuming that the model is loglinear.
        """

        if self.model.constraint is not None:
            warnings.warn("marginal effects ignore constraints",
                          ValueWarning)

        return GEEMargins(self, (at, method, atexog, dummy, count))

    def plot_isotropic_dependence(self, ax=None, xpoints=10,
                                  min_n=50):
        """
        Create a plot of the pairwise products of within-group
        residuals against the corresponding time differences.  This
        plot can be used to assess the possible form of an isotropic
        covariance structure.

        Parameters
        ----------
        ax : AxesSubplot
            An axes on which to draw the graph.  If None, new
            figure and axes objects are created
        xpoints : scalar or array_like
            If scalar, the number of points equally spaced points on
            the time difference axis used to define bins for
            calculating local means.  If an array, the specific points
            that define the bins.
        min_n : int
            The minimum sample size in a bin for the mean residual
            product to be included on the plot.
        """

        from statsmodels.graphics import utils as gutils

        resid = self.model.cluster_list(self.resid)
        time = self.model.cluster_list(self.model.time)

        # All within-group pairwise time distances (xdt) and the
        # corresponding products of scaled residuals (xre).
        xre, xdt = [], []
        for re, ti in zip(resid, time):
            ix = np.tril_indices(re.shape[0], 0)
            re = re[ix[0]] * re[ix[1]] / self.scale ** 2
            xre.append(re)
            dists = np.sqrt(((ti[ix[0], :] - ti[ix[1], :]) ** 2).sum(1))
            xdt.append(dists)

        xre = np.concatenate(xre)
        xdt = np.concatenate(xdt)

        if ax is None:
            fig, ax = gutils.create_mpl_ax(ax)
        else:
            fig = ax.get_figure()

        # Convert to a correlation
        ii = np.flatnonzero(xdt == 0)
        v0 = np.mean(xre[ii])
        xre /= v0

        # Use the simple average to smooth, since fancier smoothers
        # that trim and downweight outliers give biased results (we
        # need the actual mean of a skewed distribution).
        if np.isscalar(xpoints):
            xpoints = np.linspace(0, max(xdt), xpoints)
        dg = np.digitize(xdt, xpoints)
        dgu = np.unique(dg)
        hist = np.asarray([np.sum(dg == k) for k in dgu])
        ii = np.flatnonzero(hist >= min_n)
        dgu = dgu[ii]
        dgy = np.asarray([np.mean(xre[dg == k]) for k in dgu])
        dgx = np.asarray([np.mean(xdt[dg == k]) for k in dgu])

        ax.plot(dgx, dgy, '-', color='orange', lw=5)
        ax.set_xlabel("Time difference")
        ax.set_ylabel("Product of scaled residuals")

        return fig

    def sensitivity_params(self, dep_params_first,
                           dep_params_last, num_steps):
        """
        Refits the GEE model using a sequence of values for the
        dependence parameters.

        Parameters
        ----------
        dep_params_first : array_like
            The first dep_params in the sequence
        dep_params_last : array_like
            The last dep_params in the sequence
        num_steps : int
            The number of dep_params in the sequence

        Returns
        -------
        results : array_like
            The GEEResults objects resulting from the fits.
        """

        model = self.model

        import copy
        cov_struct = copy.deepcopy(self.model.cov_struct)

        # We are fixing the dependence structure in each run.
        update_dep = model.update_dep
        model.update_dep = False

        dep_params = []
        results = []
        for x in np.linspace(0, 1, num_steps):

            dp = x * dep_params_last + (1 - x) * dep_params_first
            dep_params.append(dp)

            model.cov_struct = copy.deepcopy(cov_struct)
            model.cov_struct.dep_params = dp
            rslt = model.fit(start_params=self.params,
                             ctol=self.ctol,
                             params_niter=self.params_niter,
                             first_dep_update=self.first_dep_update,
                             cov_type=self.cov_type)
            results.append(rslt)

        model.update_dep = update_dep

        return results

    # FIXME: alias to be removed, temporary backwards compatibility
    params_sensitivity = sensitivity_params


class GEEResultsWrapper(lm.RegressionResultsWrapper):
    _attrs = {
        'centered_resid': 'rows',
    }
    _wrap_attrs = wrap.union_dicts(lm.RegressionResultsWrapper._wrap_attrs,
                                   _attrs)
wrap.populate_wrapper(GEEResultsWrapper, GEEResults)  # noqa:E305


class OrdinalGEE(GEE):

    __doc__ = (
        "    Ordinal Response Marginal Regression Model using GEE\n" +
        _gee_init_doc % {'extra_params': base._missing_param_doc,
                         'family_doc': _gee_ordinal_family_doc,
                         'example': _gee_ordinal_example,
                         'notes': _gee_nointercept})

    def __init__(self, endog, exog, groups, time=None, family=None,
                 cov_struct=None, missing='none', offset=None,
                 dep_data=None, constraint=None, **kwargs):

        if family is None:
            family = families.Binomial()
        else:
            if not isinstance(family, families.Binomial):
                raise ValueError("ordinal GEE must use a Binomial family")

        if cov_struct is None:
            cov_struct = cov_structs.OrdinalIndependence()

        endog, exog, groups, time, offset = self.setup_ordinal(
            endog, exog, groups, time, offset)

        super().__init__(endog, exog, groups, time,
                         family, cov_struct, missing,
                         offset, dep_data, constraint)

    def setup_ordinal(self, endog, exog, groups, time, offset):
        """
        Restructure ordinal data as binary indicators so that they can
        be analyzed using Generalized Estimating Equations.
        """

        self.endog_orig = endog.copy()
        self.exog_orig = exog.copy()
        self.groups_orig = groups.copy()
        if offset is not None:
            self.offset_orig = offset.copy()
        else:
            self.offset_orig = None
            offset = np.zeros(len(endog))
        if time is not None:
            self.time_orig = time.copy()
        else:
            self.time_orig = None
            time = np.zeros((len(endog), 1))

        exog = np.asarray(exog)
        endog = np.asarray(endog)
        groups = np.asarray(groups)
        time = np.asarray(time)
        offset = np.asarray(offset)

        # The unique outcomes, except the greatest one.
        self.endog_values = np.unique(endog)
        endog_cuts = self.endog_values[0:-1]
        ncut = len(endog_cuts)

        nrows = ncut * len(endog)
        exog_out = np.zeros((nrows, exog.shape[1]),
                            dtype=np.float64)
        endog_out = np.zeros(nrows, dtype=np.float64)
        intercepts = np.zeros((nrows, ncut), dtype=np.float64)
        groups_out = np.zeros(nrows, dtype=groups.dtype)
        time_out = np.zeros((nrows, time.shape[1]),
                            dtype=np.float64)
        offset_out = np.zeros(nrows, dtype=np.float64)

        jrow = 0
        zipper = zip(exog, endog, groups, time, offset)
        for (exog_row, endog_value, group_value, time_value,
             offset_value) in zipper:

            # Loop over thresholds for the indicators
            for thresh_ix, thresh in enumerate(endog_cuts):

                exog_out[jrow, :] = exog_row
                endog_out[jrow] = int(np.squeeze(endog_value > thresh))
                intercepts[jrow, thresh_ix] = 1
                groups_out[jrow] = group_value
                time_out[jrow] = time_value
                offset_out[jrow] = offset_value
                jrow += 1

        exog_out = np.concatenate((intercepts, exog_out), axis=1)

        # exog column names, including intercepts
        xnames = ["I(y>%.1f)" % v for v in endog_cuts]
        if type(self.exog_orig) is pd.DataFrame:
            xnames.extend(self.exog_orig.columns)
        else:
            xnames.extend(["x%d" % k for k in range(1, exog.shape[1] + 1)])
        exog_out = pd.DataFrame(exog_out, columns=xnames)

        # Preserve the endog name if there is one
        if type(self.endog_orig) is pd.Series:
            endog_out = pd.Series(endog_out, name=self.endog_orig.name)

        return endog_out, exog_out, groups_out, time_out, offset_out

    def _starting_params(self):
        exposure = getattr(self, "exposure", None)
        model = GEE(self.endog, self.exog, self.groups,
                    time=self.time, family=families.Binomial(),
                    offset=self.offset, exposure=exposure)
        result = model.fit()
        return result.params

    @Appender(_gee_fit_doc)
    def fit(self, maxiter=60, ctol=1e-6, start_params=None,
            params_niter=1, first_dep_update=0,
            cov_type='robust'):

        rslt = super().fit(maxiter, ctol, start_params,
                           params_niter, first_dep_update,
                           cov_type=cov_type)

        rslt = rslt._results   # use unwrapped instance
        res_kwds = {k: getattr(rslt, k) for k in rslt._props}
        # Convert the GEEResults to an OrdinalGEEResults
        ord_rslt = OrdinalGEEResults(self, rslt.params,
                                     rslt.cov_params() / rslt.scale,
                                     rslt.scale,
                                     cov_type=cov_type,
                                     attr_kwds=res_kwds)
        # for k in rslt._props:
        #    setattr(ord_rslt, k, getattr(rslt, k))
        # TODO: document or delete

        return OrdinalGEEResultsWrapper(ord_rslt)


class OrdinalGEEResults(GEEResults):

    __doc__ = (
        "This class summarizes the fit of a marginal regression model"
        "for an ordinal response using GEE.\n"
        + _gee_results_doc)

    def plot_distribution(self, ax=None, exog_values=None):
        """
        Plot the fitted probabilities of endog in an ordinal model,
        for specified values of the predictors.

        Parameters
        ----------
        ax : AxesSubplot
            An axes on which to draw the graph.  If None, new
            figure and axes objects are created
        exog_values : array_like
            A list of dictionaries, with each dictionary mapping
            variable names to values at which the variable is held
            fixed.  The values P(endog=y | exog) are plotted for all
            possible values of y, at the given exog value.  Variables
            not included in a dictionary are held fixed at the mean
            value.

        Example:
        --------
        We have a model with covariates 'age' and 'sex', and wish to
        plot the probabilities P(endog=y | exog) for males (sex=0) and
        for females (sex=1), as separate paths on the plot.  Since
        'age' is not included below in the map, it is held fixed at
        its mean value.

        >>> ev = [{"sex": 1}, {"sex": 0}]
        >>> rslt.distribution_plot(exog_values=ev)
        """

        from statsmodels.graphics import utils as gutils

        if ax is None:
            fig, ax = gutils.create_mpl_ax(ax)
        else:
            fig = ax.get_figure()

        # If no covariate patterns are specified, create one with all
        # variables set to their mean values.
        if exog_values is None:
            exog_values = [{}, ]

        exog_means = self.model.exog.mean(0)
        ix_icept = [i for i, x in enumerate(self.model.exog_names) if
                    x.startswith("I(")]

        for ev in exog_values:

            for k in ev.keys():
                if k not in self.model.exog_names:
                    raise ValueError("%s is not a variable in the model"
                                     % k)

            # Get the fitted probability for each level, at the given
            # covariate values.
            pr = []
            for j in ix_icept:

                xp = np.zeros_like(self.params)
                xp[j] = 1.
                for i, vn in enumerate(self.model.exog_names):
                    if i in ix_icept:
                        continue
                    # User-specified value
                    if vn in ev:
                        xp[i] = ev[vn]
                    # Mean value
                    else:
                        xp[i] = exog_means[i]

                p = 1 / (1 + np.exp(-np.dot(xp, self.params)))
                pr.append(p)

            pr.insert(0, 1)
            pr.append(0)
            pr = np.asarray(pr)
            prd = -np.diff(pr)

            ax.plot(self.model.endog_values, prd, 'o-')

        ax.set_xlabel("Response value")
        ax.set_ylabel("Probability")
        ax.set_ylim(0, 1)

        return fig


def _score_test_submodel(par, sub):
    """
    Return transformation matrices for design matrices.

    Parameters
    ----------
    par : instance
        The parent model
    sub : instance
        The sub-model

    Returns
    -------
    qm : array_like
        Matrix mapping the design matrix of the parent to the design matrix
        for the sub-model.
    qc : array_like
        Matrix mapping the design matrix of the parent to the orthogonal
        complement of the columnspace of the submodel in the columnspace
        of the parent.

    Notes
    -----
    Returns None, None if the provided submodel is not actually a submodel.
    """

    x1 = par.exog
    x2 = sub.exog

    u, s, vt = np.linalg.svd(x1, 0)
    v = vt.T

    # Get the orthogonal complement of col(x2) in col(x1).
    a, _ = np.linalg.qr(x2)
    a = u - np.dot(a, np.dot(a.T, u))
    x2c, sb, _ = np.linalg.svd(a, 0)
    x2c = x2c[:, sb > 1e-12]

    # x1 * qm = x2
    ii = np.flatnonzero(np.abs(s) > 1e-12)
    qm = np.dot(v[:, ii], np.dot(u[:, ii].T, x2) / s[ii, None])

    e = np.max(np.abs(x2 - np.dot(x1, qm)))
    if e > 1e-8:
        return None, None

    # x1 * qc = x2c
    qc = np.dot(v[:, ii], np.dot(u[:, ii].T, x2c) / s[ii, None])

    return qm, qc


class OrdinalGEEResultsWrapper(GEEResultsWrapper):
    pass
wrap.populate_wrapper(OrdinalGEEResultsWrapper, OrdinalGEEResults)  # noqa:E305


class NominalGEE(GEE):

    __doc__ = (
        "    Nominal Response Marginal Regression Model using GEE.\n" +
        _gee_init_doc % {'extra_params': base._missing_param_doc,
                         'family_doc': _gee_nominal_family_doc,
                         'example': _gee_nominal_example,
                         'notes': _gee_nointercept})

    def __init__(self, endog, exog, groups, time=None, family=None,
                 cov_struct=None, missing='none', offset=None,
                 dep_data=None, constraint=None, **kwargs):

        endog, exog, groups, time, offset = self.setup_nominal(
            endog, exog, groups, time, offset)

        if family is None:
            family = _Multinomial(self.ncut + 1)

        if cov_struct is None:
            cov_struct = cov_structs.NominalIndependence()

        super().__init__(
            endog, exog, groups, time, family, cov_struct, missing,
            offset, dep_data, constraint)

    def _starting_params(self):
        exposure = getattr(self, "exposure", None)
        model = GEE(self.endog, self.exog, self.groups,
                    time=self.time, family=families.Binomial(),
                    offset=self.offset, exposure=exposure)
        result = model.fit()
        return result.params

    def setup_nominal(self, endog, exog, groups, time, offset):
        """
        Restructure nominal data as binary indicators so that they can
        be analyzed using Generalized Estimating Equations.
        """

        self.endog_orig = endog.copy()
        self.exog_orig = exog.copy()
        self.groups_orig = groups.copy()
        if offset is not None:
            self.offset_orig = offset.copy()
        else:
            self.offset_orig = None
            offset = np.zeros(len(endog))
        if time is not None:
            self.time_orig = time.copy()
        else:
            self.time_orig = None
            time = np.zeros((len(endog), 1))

        exog = np.asarray(exog)
        endog = np.asarray(endog)
        groups = np.asarray(groups)
        time = np.asarray(time)
        offset = np.asarray(offset)

        # The unique outcomes, except the greatest one.
        self.endog_values = np.unique(endog)
        endog_cuts = self.endog_values[0:-1]
        ncut = len(endog_cuts)
        self.ncut = ncut

        nrows = len(endog_cuts) * exog.shape[0]
        ncols = len(endog_cuts) * exog.shape[1]
        exog_out = np.zeros((nrows, ncols), dtype=np.float64)
        endog_out = np.zeros(nrows, dtype=np.float64)
        groups_out = np.zeros(nrows, dtype=np.float64)
        time_out = np.zeros((nrows, time.shape[1]),
                            dtype=np.float64)
        offset_out = np.zeros(nrows, dtype=np.float64)

        jrow = 0
        zipper = zip(exog, endog, groups, time, offset)
        for (exog_row, endog_value, group_value, time_value,
             offset_value) in zipper:

            # Loop over thresholds for the indicators
            for thresh_ix, thresh in enumerate(endog_cuts):

                u = np.zeros(len(endog_cuts), dtype=np.float64)
                u[thresh_ix] = 1
                exog_out[jrow, :] = np.kron(u, exog_row)
                endog_out[jrow] = int(np.squeeze(endog_value == thresh))
                groups_out[jrow] = group_value
                time_out[jrow] = time_value
                offset_out[jrow] = offset_value
                jrow += 1

        # exog names
        if isinstance(self.exog_orig, pd.DataFrame):
            xnames_in = self.exog_orig.columns
        else:
            xnames_in = ["x%d" % k for k in range(1, exog.shape[1] + 1)]
        xnames = []
        for tr in endog_cuts:
            xnames.extend([f"{v}[{tr:.1f}]" for v in xnames_in])
        exog_out = pd.DataFrame(exog_out, columns=xnames)
        exog_out = pd.DataFrame(exog_out, columns=xnames)

        # Preserve endog name if there is one
        if isinstance(self.endog_orig, pd.Series):
            endog_out = pd.Series(endog_out, name=self.endog_orig.name)

        return endog_out, exog_out, groups_out, time_out, offset_out

    def mean_deriv(self, exog, lin_pred):
        """
        Derivative of the expected endog with respect to the parameters.

        Parameters
        ----------
        exog : array_like
           The exogeneous data at which the derivative is computed,
           number of rows must be a multiple of `ncut`.
        lin_pred : array_like
           The values of the linear predictor, length must be multiple
           of `ncut`.

        Returns
        -------
        The derivative of the expected endog with respect to the
        parameters.
        """

        expval = np.exp(lin_pred)

        # Reshape so that each row contains all the indicators
        # corresponding to one multinomial observation.
        expval_m = np.reshape(expval, (len(expval) // self.ncut,
                                       self.ncut))

        # The normalizing constant for the multinomial probabilities.
        denom = 1 + expval_m.sum(1)
        denom = np.kron(denom, np.ones(self.ncut, dtype=np.float64))

        # The multinomial probabilities
        mprob = expval / denom

        # First term of the derivative: denom * expval' / denom^2 =
        # expval' / denom.
        dmat = mprob[:, None] * exog

        # Second term of the derivative: -expval * denom' / denom^2
        ddenom = expval[:, None] * exog
        dmat -= mprob[:, None] * ddenom / denom[:, None]

        return dmat

    def mean_deriv_exog(self, exog, params, offset_exposure=None):
        """
        Derivative of the expected endog with respect to exog for the
        multinomial model, used in analyzing marginal effects.

        Parameters
        ----------
        exog : array_like
           The exogeneous data at which the derivative is computed,
           number of rows must be a multiple of `ncut`.
        lpr : array_like
           The linear predictor values, length must be multiple of
           `ncut`.

        Returns
        -------
        The value of the derivative of the expected endog with respect
        to exog.

        Notes
        -----
        offset_exposure must be set at None for the multinomial family.
        """

        if offset_exposure is not None:
            warnings.warn("Offset/exposure ignored for the multinomial family",
                          ValueWarning)

        lpr = np.dot(exog, params)
        expval = np.exp(lpr)

        expval_m = np.reshape(expval, (len(expval) // self.ncut,
                                       self.ncut))

        denom = 1 + expval_m.sum(1)
        denom = np.kron(denom, np.ones(self.ncut, dtype=np.float64))

        bmat0 = np.outer(np.ones(exog.shape[0]), params)

        # Masking matrix
        qmat = []
        for j in range(self.ncut):
            ee = np.zeros(self.ncut, dtype=np.float64)
            ee[j] = 1
            qmat.append(np.kron(ee, np.ones(len(params) // self.ncut)))
        qmat = np.array(qmat)
        qmat = np.kron(np.ones((exog.shape[0] // self.ncut, 1)), qmat)
        bmat = bmat0 * qmat

        dmat = expval[:, None] * bmat / denom[:, None]

        expval_mb = np.kron(expval_m, np.ones((self.ncut, 1)))
        expval_mb = np.kron(expval_mb, np.ones((1, self.ncut)))

        dmat -= expval[:, None] * (bmat * expval_mb) / denom[:, None] ** 2

        return dmat

    @Appender(_gee_fit_doc)
    def fit(self, maxiter=60, ctol=1e-6, start_params=None,
            params_niter=1, first_dep_update=0,
            cov_type='robust'):

        rslt = super().fit(maxiter, ctol, start_params,
                           params_niter, first_dep_update,
                           cov_type=cov_type)
        if rslt is None:
            warnings.warn("GEE updates did not converge",
                          ConvergenceWarning)
            return None

        rslt = rslt._results   # use unwrapped instance
        res_kwds = {k: getattr(rslt, k) for k in rslt._props}
        # Convert the GEEResults to a NominalGEEResults
        nom_rslt = NominalGEEResults(self, rslt.params,
                                     rslt.cov_params() / rslt.scale,
                                     rslt.scale,
                                     cov_type=cov_type,
                                     attr_kwds=res_kwds)
        # TODO: document or delete
        # for k in rslt._props:
        #    setattr(nom_rslt, k, getattr(rslt, k))

        return NominalGEEResultsWrapper(nom_rslt)


class NominalGEEResults(GEEResults):

    __doc__ = (
        "This class summarizes the fit of a marginal regression model"
        "for a nominal response using GEE.\n"
        + _gee_results_doc)

    def plot_distribution(self, ax=None, exog_values=None):
        """
        Plot the fitted probabilities of endog in an nominal model,
        for specified values of the predictors.

        Parameters
        ----------
        ax : AxesSubplot
            An axes on which to draw the graph.  If None, new
            figure and axes objects are created
        exog_values : array_like
            A list of dictionaries, with each dictionary mapping
            variable names to values at which the variable is held
            fixed.  The values P(endog=y | exog) are plotted for all
            possible values of y, at the given exog value.  Variables
            not included in a dictionary are held fixed at the mean
            value.

        Example:
        --------
        We have a model with covariates 'age' and 'sex', and wish to
        plot the probabilities P(endog=y | exog) for males (sex=0) and
        for females (sex=1), as separate paths on the plot.  Since
        'age' is not included below in the map, it is held fixed at
        its mean value.

        >>> ex = [{"sex": 1}, {"sex": 0}]
        >>> rslt.distribution_plot(exog_values=ex)
        """

        from statsmodels.graphics import utils as gutils

        if ax is None:
            fig, ax = gutils.create_mpl_ax(ax)
        else:
            fig = ax.get_figure()

        # If no covariate patterns are specified, create one with all
        # variables set to their mean values.
        if exog_values is None:
            exog_values = [{}, ]

        link = self.model.family.link.inverse
        ncut = self.model.family.ncut

        k = int(self.model.exog.shape[1] / ncut)
        exog_means = self.model.exog.mean(0)[0:k]
        exog_names = self.model.exog_names[0:k]
        exog_names = [x.split("[")[0] for x in exog_names]

        params = np.reshape(self.params,
                            (ncut, len(self.params) // ncut))

        for ev in exog_values:

            exog = exog_means.copy()

            for k in ev.keys():
                if k not in exog_names:
                    raise ValueError("%s is not a variable in the model"
                                     % k)

                ii = exog_names.index(k)
                exog[ii] = ev[k]

            lpr = np.dot(params, exog)
            pr = link(lpr)
            pr = np.r_[pr, 1 - pr.sum()]

            ax.plot(self.model.endog_values, pr, 'o-')

        ax.set_xlabel("Response value")
        ax.set_ylabel("Probability")
        ax.set_xticks(self.model.endog_values)
        ax.set_xticklabels(self.model.endog_values)
        ax.set_ylim(0, 1)

        return fig


class NominalGEEResultsWrapper(GEEResultsWrapper):
    pass
wrap.populate_wrapper(NominalGEEResultsWrapper, NominalGEEResults)  # noqa:E305


class _MultinomialLogit(Link):
    """
    The multinomial logit transform, only for use with GEE.

    Notes
    -----
    The data are assumed coded as binary indicators, where each
    observed multinomial value y is coded as I(y == S[0]), ..., I(y ==
    S[-1]), where S is the set of possible response labels, excluding
    the largest one.  Thererefore functions in this class should only
    be called using vector argument whose length is a multiple of |S|
    = ncut, which is an argument to be provided when initializing the
    class.

    call and derivative use a private method _clean to trim p by 1e-10
    so that p is in (0, 1)
    """

    def __init__(self, ncut):
        self.ncut = ncut

    def inverse(self, lpr):
        """
        Inverse of the multinomial logit transform, which gives the
        expected values of the data as a function of the linear
        predictors.

        Parameters
        ----------
        lpr : array_like (length must be divisible by `ncut`)
            The linear predictors

        Returns
        -------
        prob : ndarray
            Probabilities, or expected values
        """

        expval = np.exp(lpr)

        denom = 1 + np.reshape(expval, (len(expval) // self.ncut,
                                        self.ncut)).sum(1)
        denom = np.kron(denom, np.ones(self.ncut, dtype=np.float64))

        prob = expval / denom

        return prob


class _Multinomial(families.Family):
    """
    Pseudo-link function for fitting nominal multinomial models with
    GEE.  Not for use outside the GEE class.
    """

    links = [_MultinomialLogit, ]
    variance = varfuncs.binary
    safe_links = [_MultinomialLogit, ]

    def __init__(self, nlevels, check_link=True):
        """
        Parameters
        ----------
        nlevels : int
            The number of distinct categories for the multinomial
            distribution.
        """
        self._check_link = check_link
        self.initialize(nlevels)

    def initialize(self, nlevels):
        self.ncut = nlevels - 1
        self.link = _MultinomialLogit(self.ncut)


class GEEMargins:
    """
    Estimated marginal effects for a regression model fit with GEE.

    Parameters
    ----------
    results : GEEResults instance
        The results instance of a fitted discrete choice model
    args : tuple
        Args are passed to `get_margeff`. This is the same as
        results.get_margeff. See there for more information.
    kwargs : dict
        Keyword args are passed to `get_margeff`. This is the same as
        results.get_margeff. See there for more information.
    """

    def __init__(self, results, args, kwargs={}):
        self._cache = {}
        self.results = results
        self.get_margeff(*args, **kwargs)

    def _reset(self):
        self._cache = {}

    @cache_readonly
    def tvalues(self):
        _check_at_is_all(self.margeff_options)
        return self.margeff / self.margeff_se

    def summary_frame(self, alpha=.05):
        """
        Returns a DataFrame summarizing the marginal effects.

        Parameters
        ----------
        alpha : float
            Number between 0 and 1. The confidence intervals have the
            probability 1-alpha.

        Returns
        -------
        frame : DataFrames
            A DataFrame summarizing the marginal effects.
        """
        _check_at_is_all(self.margeff_options)
        from pandas import DataFrame
        names = [_transform_names[self.margeff_options['method']],
                 'Std. Err.', 'z', 'Pr(>|z|)',
                 'Conf. Int. Low', 'Cont. Int. Hi.']
        ind = self.results.model.exog.var(0) != 0  # True if not a constant
        exog_names = self.results.model.exog_names
        var_names = [name for i, name in enumerate(exog_names) if ind[i]]
        table = np.column_stack((self.margeff, self.margeff_se, self.tvalues,
                                 self.pvalues, self.conf_int(alpha)))
        return DataFrame(table, columns=names, index=var_names)

    @cache_readonly
    def pvalues(self):
        _check_at_is_all(self.margeff_options)
        return stats.norm.sf(np.abs(self.tvalues)) * 2

    def conf_int(self, alpha=.05):
        """
        Returns the confidence intervals of the marginal effects

        Parameters
        ----------
        alpha : float
            Number between 0 and 1. The confidence intervals have the
            probability 1-alpha.

        Returns
        -------
        conf_int : ndarray
            An array with lower, upper confidence intervals for the marginal
            effects.
        """
        _check_at_is_all(self.margeff_options)
        me_se = self.margeff_se
        q = stats.norm.ppf(1 - alpha / 2)
        lower = self.margeff - q * me_se
        upper = self.margeff + q * me_se
        return np.asarray(lzip(lower, upper))

    def summary(self, alpha=.05):
        """
        Returns a summary table for marginal effects

        Parameters
        ----------
        alpha : float
            Number between 0 and 1. The confidence intervals have the
            probability 1-alpha.

        Returns
        -------
        Summary : SummaryTable
            A SummaryTable instance
        """
        _check_at_is_all(self.margeff_options)
        results = self.results
        model = results.model
        title = model.__class__.__name__ + " Marginal Effects"
        method = self.margeff_options['method']
        top_left = [('Dep. Variable:', [model.endog_names]),
                    ('Method:', [method]),
                    ('At:', [self.margeff_options['at']]), ]

        from statsmodels.iolib.summary import (Summary, summary_params,
                                               table_extend)
        exog_names = model.exog_names[:]  # copy
        smry = Summary()

        const_idx = model.data.const_idx
        if const_idx is not None:
            exog_names.pop(const_idx)

        J = int(getattr(model, "J", 1))
        if J > 1:
            yname, yname_list = results._get_endog_name(model.endog_names,
                                                        None, all=True)
        else:
            yname = model.endog_names
            yname_list = [yname]

        smry.add_table_2cols(self, gleft=top_left, gright=[],
                             yname=yname, xname=exog_names, title=title)

        # NOTE: add_table_params is not general enough yet for margeff
        # could use a refactor with getattr instead of hard-coded params
        # tvalues etc.
        table = []
        conf_int = self.conf_int(alpha)
        margeff = self.margeff
        margeff_se = self.margeff_se
        tvalues = self.tvalues
        pvalues = self.pvalues
        if J > 1:
            for eq in range(J):
                restup = (results, margeff[:, eq], margeff_se[:, eq],
                          tvalues[:, eq], pvalues[:, eq], conf_int[:, :, eq])
                tble = summary_params(restup, yname=yname_list[eq],
                                      xname=exog_names, alpha=alpha,
                                      use_t=False,
                                      skip_header=True)
                tble.title = yname_list[eq]
                # overwrite coef with method name
                header = ['', _transform_names[method], 'std err', 'z',
                          'P>|z|',
                          '[%3.1f%% Conf. Int.]' % (100 - alpha * 100)]
                tble.insert_header_row(0, header)
                # from IPython.core.debugger import Pdb; Pdb().set_trace()
                table.append(tble)

            table = table_extend(table, keep_headers=True)
        else:
            restup = (results, margeff, margeff_se, tvalues, pvalues, conf_int)
            table = summary_params(restup, yname=yname, xname=exog_names,
                                   alpha=alpha, use_t=False, skip_header=True)
            header = ['', _transform_names[method], 'std err', 'z',
                      'P>|z|', '[%3.1f%% Conf. Int.]' % (100 - alpha * 100)]
            table.insert_header_row(0, header)

        smry.tables.append(table)
        return smry

    def get_margeff(self, at='overall', method='dydx', atexog=None,
                    dummy=False, count=False):

        self._reset()  # always reset the cache when this is called
        # TODO: if at is not all or overall, we can also put atexog values
        # in summary table head
        method = method.lower()
        at = at.lower()
        _check_margeff_args(at, method)
        self.margeff_options = dict(method=method, at=at)
        results = self.results
        model = results.model
        params = results.params
        exog = model.exog.copy()  # copy because values are changed
        effects_idx = exog.var(0) != 0
        const_idx = model.data.const_idx

        if dummy:
            _check_discrete_args(at, method)
            dummy_idx, dummy = _get_dummy_index(exog, const_idx)
        else:
            dummy_idx = None

        if count:
            _check_discrete_args(at, method)
            count_idx, count = _get_count_index(exog, const_idx)
        else:
            count_idx = None

        # get the exogenous variables
        exog = _get_margeff_exog(exog, at, atexog, effects_idx)

        # get base marginal effects, handled by sub-classes
        effects = model._derivative_exog(params, exog, method,
                                         dummy_idx, count_idx)
        effects = _effects_at(effects, at)

        if at == 'all':
            self.margeff = effects[:, effects_idx]
        else:
            # Set standard error of the marginal effects by Delta method.
            margeff_cov, margeff_se = margeff_cov_with_se(
                model, params, exog, results.cov_params(), at,
                model._derivative_exog, dummy_idx, count_idx,
                method, 1)

            # do not care about at constant
            self.margeff_cov = margeff_cov[effects_idx][:, effects_idx]
            self.margeff_se = margeff_se[effects_idx]
            self.margeff = effects[effects_idx]
